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OPTICAL WAVEFRONT RECONSTRUCTION: THEORY AND NUMERICAL METHODS
"... Optical wavefront reconstruction algorithms played a central role in the effort to identify gross manufacturing errors in NASA's Hubble Space Telescope (HST). NASA's success with reconstruction algorithms on the HST has lead to an effort to develop software that can aid and in some cases ..."
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Cited by 28 (10 self)
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Optical wavefront reconstruction algorithms played a central role in the effort to identify gross manufacturing errors in NASA's Hubble Space Telescope (HST). NASA's success with reconstruction algorithms on the HST has lead to an effort to develop software that can aid and in some cases replace complicated, expensive and errorprone hardware. Among the many applications is HST's replacement, the Next Generation Space Telescope (NGST). This work details the theory of optical wavefront reconstruction, reviews some numerical methods for this problem, and presents a novel numerical technique which we call extended least squares. We compare the performance of these numerical methods for potential inclusion in prototype NGST optical wavefront reconstruction software. We begin with a tutorial of RayleighSommerfeld diffraction theory.
Numerical methods for coupled superresolution
 Inverse Probl
, 2006
"... The process of combining, via mathematical software tools, a set of low resolution images into a single high resolution image is often referred to as superresolution. Algorithms for superresolution involve two key steps: registration and reconstruction. Most approaches proposed in the literature d ..."
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Cited by 19 (5 self)
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The process of combining, via mathematical software tools, a set of low resolution images into a single high resolution image is often referred to as superresolution. Algorithms for superresolution involve two key steps: registration and reconstruction. Most approaches proposed in the literature decouple these steps, solving each independently. This can be effective if there are very simple, linear displacements between the low resolution images. However, for more complex, nonlinear, nonuniform transformations, estimating the displacements can be very difficult, leading to severe inaccuracies in the reconstructed high resolution image. This paper presents a mathematical framework and optimization algorithms that can be used to jointly estimate these quantities. Efficient implementation details are considered, and numerical experiments are provided to illustrate the effectiveness of our approach.
Blind iterative restoration of images with spatiallyvarying blur
 In Optics Express
, 2006
"... Removing nonuniform blur and noise from optical images is a very difficult problem to resolve. In this paper we describe a strategy that can be used for solving such problems. We describe how to restore images blurred by an unknown spatiallyvarying point spread function (PSF) by using a combinatio ..."
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Cited by 17 (0 self)
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Removing nonuniform blur and noise from optical images is a very difficult problem to resolve. In this paper we describe a strategy that can be used for solving such problems. We describe how to restore images blurred by an unknown spatiallyvarying point spread function (PSF) by using a combination of methods including sectioning and phase diversity blind deconvolution. The PSFs on the individual sections are not known in advance. We treat the sections as a sequence of frames whose PSFs are correlated and approximately spatiallyinvariant, and apply iterative blind deconvolution schemes based on phase diversity to approximate these PSFs. A technique by Nagy and O’Leary is then used to restore the image globally. Test results on star cluster data are presented. 1.
MultiFrame Blind Deconvolution With Linear Equality Constraints
 Proc. SPIE 2002, 4792, 146155.  12  Intensity 1.0 0.8 0.6 0.4 0.2 0.0 10 5 0 5 10 Position in Image Plane
, 2002
"... The Phase Diverse Speckle (PDS) problem is formulated mathematically as Multi Frame Blind Deconvolution (MFBD) together with a set of Linear Equality Constraints (LECs) on the wavefront expansion parameters. This MFBDLEC formulation is quite general and, in addition to PDS, it allows the same code ..."
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Cited by 7 (2 self)
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The Phase Diverse Speckle (PDS) problem is formulated mathematically as Multi Frame Blind Deconvolution (MFBD) together with a set of Linear Equality Constraints (LECs) on the wavefront expansion parameters. This MFBDLEC formulation is quite general and, in addition to PDS, it allows the same code to handle a variety of different data collection schemes specified as data, the LECs, rather than in the code. It also relieves us from having to derive new expressions for the gradient of the wavefront parameter vector for each type of data set. The idea is first presented with a simple formulation that accommodates Phase Diversity, Phase Diverse Speckle, and ShackHartmann wavefront sensing. Then various generalizations are discussed, that allows many other types of data sets to be handled.
Fast Phase Diversity Wavefront Sensing for Mirror Control
 in Adaptive Optical System Technologies, D. Bonnaccini and
, 1998
"... We show with simulation experiments that closedloop phasediversity can be used without numerical guardbands for wavefront sensing of loworder wavefronts from extended objects using broadband filters. This may allow realtime correction at high bandwidth for certain applications. We also present ..."
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We show with simulation experiments that closedloop phasediversity can be used without numerical guardbands for wavefront sensing of loworder wavefronts from extended objects using broadband filters. This may allow realtime correction at high bandwidth for certain applications. We also present a proper maximum likelihood treatment of ShackHartman data, which includes an imaging model to extract curvature information from the lenslet images. We demonstrate by simple simulations that this approach should allow higherorder wavefront information to be extracted than with with traditional ShackHartmann wavefront sensing for a given number of lenslets.
Regularization Methods For Blind Deconvolution And Blind Source Separation Problems
 Math. Cont. Signals & Systems
, 2001
"... . This paper is devoted to blind deconvolution and blind separation problems. Blind deconvolution is the identification of a point spread function and an input signal from an observation of their convolution. Blind source separation is the recovery of a vector of input signals from a vector of obser ..."
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. This paper is devoted to blind deconvolution and blind separation problems. Blind deconvolution is the identification of a point spread function and an input signal from an observation of their convolution. Blind source separation is the recovery of a vector of input signals from a vector of observed signals, which are mixed by a linear (unknown) operator. We show that both problems are paradigms of nonlinear illposed problems. Consequently, regularization techniques have to be used for stable numerical reconstructions. In this paper we develop a rigorous convergence analysis for regularization techniques for the solution of blind deconvolution and blind separation problems. We prove convergence of the alternating minimization algorithm for the numerical solution of regularized blind deconvolution problems and present some numerical examples. Moreover, we show that many neural network approaches for blind inversion can be considered in the framework of regularization theory. Keywords: Alternating Minimization Algorithm, Blind Deconvolution, Blind Source Separation, IllPosed Problems, Neural Networks, Regularization, Signal and Image Processing. AMS Subject Classification: 35J15, 60G35, 65J20, 65K10, 68U10, 92B20. Abbreviated Title: Regularization Methods for Blind Deconvolution. 1.
Blind Deconvolution and Structured Matrix Computations with Applications to Array Imaging,” Blind Deconvolution: Theory and Applications
, 2007
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Statistical behavior of joint least square estimation in the phase diversity context
 IEEE Trans. Image Processing
, 2005
"... Abstract—The images recorded by optical telescopes are often degraded by aberrations that induce phase variations in the pupil plane. Several wavefront sensing techniques have been proposed to estimate aberrated phases. One of them is phase diversity, for which the joint leastsquare approach introd ..."
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Abstract—The images recorded by optical telescopes are often degraded by aberrations that induce phase variations in the pupil plane. Several wavefront sensing techniques have been proposed to estimate aberrated phases. One of them is phase diversity, for which the joint leastsquare approach introduced by Gonsalves et al. is a reference method to estimate phase coefficients from the recorded images. In this paper, we rely on the asymptotic theory of Toeplitz matrices to show that Gonsalves ’ technique provides a consistent phase estimator as the size of the images grows. No comparable result is yielded by the classical joint maximum likelihood interpretation (e.g., as found in the work by Paxman et al.). Finally, our theoretical analysis is illustrated through simulated problems. Index Terms—Error analysis, leastsquares methods, optical image processing, parameter estimation, phase diversity, statistics, Toeplitz matrices. I.
A nonnegatively constrained trust region algorithm for the restoration of images with unknown blur, Electron
 Trans. Numer. Anal
"... Abstract. We consider a largescale optimization problem with nonnegativity constraints that arises in an application of phase diversity to astronomical imaging. We develop a cost function that incorporates information about the statistics of atmospheric turbulence, and we use Tikhonov regularizatio ..."
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Abstract. We consider a largescale optimization problem with nonnegativity constraints that arises in an application of phase diversity to astronomical imaging. We develop a cost function that incorporates information about the statistics of atmospheric turbulence, and we use Tikhonov regularization to induce stability. We introduce an efficient and easily implementable algorithm that intersperses gradient projection iterations with iterations from a wellknown, unconstrained Newton/trust region method. Due to the large size of our problem and to the fact that our cost function is not convex, we approximately solve the trust region subproblem via the SteihaugToint truncated CG iteration. Iterations from the trust region algorithm are restricted to the inactive variables. We also present a highly effective preconditioner that dramatically speeds up the convergence of our algorithm. A numerical comparison using real data between our method and another standard largescale, bound constrained optimization algorithm is presented.